We present methods for computing the distance from a Boolean polynomial on$m$variables of degree$m-3$(i.e., a member of the Reed–Muller code$RM(m-3, m)$) to the space of lower-degree polynomials ($RM(m-4, m)$). The methods give verifiable certificates for both the lower and upper bounds on this distance. By applying these methods to representative lists of polynomials, we show that the covering radius of$RM(4,8)$in$RM(5,8)$is 26 and the covering radius of$RM(5,9)$in$RM(6,9)$is between 28 and 32 inclusive, and we get improved lower bounds for higher$m$. We also apply our methods to various polynomials in the literature, thereby improving the known bounds on the distance from 2-resilient polynomials to$RM(m-4, m)$.

Determining the achievable rate region for networks using routing, linear coding, or nonlinear coding is thought to be a difficult task in general, and few are known. We describe the achievable rate regions for four interesting networks (completely for three and partially for the fourth). In addition to the known matrix-computation method for proving outer bounds for linear coding, we present a new method that yields actual characteristic-dependent linear rank inequalities from which the desired bounds follow immediately.

Two characteristic-dependent linear rank inequalities are given for eight variables. In particular, the first inequality holds for all finite fields whose characteristic is not three and does not in general hold over characteristic three. The second inequality holds for all finite fields whose characteristic is three and does not in general hold over characteristics other than three. Applications of these inequalities to the computation of capacity upper bounds in network coding are demonstrated.

It is known that information inequalities on four random variables cannot be generated from a finite list. For the analogous case of linear rank variables, it is known that they can be generated from a finite list for up to five variables, but this is not known for six or more variables. Here we present partial results of computations on six-variable linear rank inequalities, showing that the number of sharp inequalities (those which cannot be generated from other inequalities) is more than one billion (counting variable-permuted forms). The problem is too large for standard polytope computation software; we describe the techniques used to generate and verify the current list of inequalities and a correspondingly large list of representable polymatroids. We also describe observed properties of the inequalities (some of which are now proven general results).

Two characteristic-dependent linear rank inequalities are given for eight variables. Specifically, the first inequality holds for all finite fields whose characteristic is not three and does not in general hold over characteristic three. The second inequality holds for all finite fields whose characteristic is three and does not in general hold over characteristics other than three. Applications of these inequalities to the computation of capacity upper bounds in network coding are demonstrated.

Networks derived from matroids have played a fundamental role in proving theoretical results about the limits of network coding. In this tutorial paper, we review many connections between matroids and network coding theory, with specific emphasis on network solvability, admissible network alphabet sizes, linear coding, and network capacity.

If beta and gamma are nonnegative integers and F is a field, then a polynomial collection {p1,..., pbeta}subeZ[alpha1,..., alphagamma] is said to be solvable over F if there exist omega1,..., omegagammaisinF such that for all i=1,..., beta we have pi(omega1,..., omegagamma)=0. We say that a network and a polynomial collection are solvably equivalent if for each field F the network has a scalar-linear solution over F if and only if the polynomial collection is solvable over F. Koetter and Medardpsilas work implies that for any directed acyclic network, there exists a solvably equivalent polynomial collection. We provide the converse result, namely that for any polynomial collection there exists a solvably equivalent directed acyclic network. (Hence, the problems of network scalar-linear solvability and polynomial collection solvability have the same complexity.) The construction of the network is modeled on a matroid construction using finite projective planes, due to MacLane in 1936. A set psi of prime numbers is a set of characteristics of a network if for every qisinpsi, the network has a scalar-linear solution over some finite field with characteristic q and does not have a scalar-linear solution over any finite field whose characteristic lies outside of psi. We show that a collection of primes is a set of characteristics of some network if and only if the collection is finite or co-finite. Two networks N and N' are ls-equivalent if for any finite field F, N is scalar-linearly solvable over F if and only if N' is scalar-linearly solvable over F. We further show that every network is ls-equivalent to a multiple-unicast matroidal network.

If beta and gamma are nonnegative integers and F is a field, then a polynomial collection {p1,hellip,Pbeta} sube Z[alpha1,hellip, alphagamma] is said to be solvable over F if there exist omega1hellip, omegagammaisin F such that for all i = 1,hellip, beta we have pi(omega1hellip, omegagamma) = 0. We say that a network and a polynomial collection are solvably equivalent if for each field F the network has a scalar-linear solution over F if and only if the polynomial collection is solvable over F. Koetter and Medard's work implies that for any directed acyclic network, there exists a solvably equivalent polynomial collection. We provide the converse result, namely, that for any polynomial collection there exists a solvably equivalent directed acyclic network. (Hence, the problems of network scalar-linear solvability and polynomial collection solvability have the same complexity.) The construction of the network is modeled on a matroid construction using finite projective planes, due to MacLane in 1936. A set psi of prime numbers is a set of characteristics of a network if for every q isin psi, the network has a scalar-linear solution over some finite field with characteristic q and does not have a scalar-linear solution over any finite field whose characteristic lies outside of psi. We show that a collection of primes is a set of characteristics of some network if and only if the collection is finite or co-finite. Two networks N and N' are Is-equivalent if for any finite field F, N is scalar-linearly solvable over F if and only if N' is scalar- linearly solvable over F. We further show that every network is ls-equivalent to a multiple-unicast matroidal network.

We define a class of networks, called matroidal networks, which includes as special cases all scalar-linearly solvable networks, and in particular solvable multicast networks. We then present a method for constructing matroidal networks from known matroids. We specifically construct networks that play an important role in proving results in the literature, such as the insufficiency of linear network coding and the unachievability of network coding capacity. We also construct a new network, from the Vamos matroid, which we call the Vamos network, and use it to prove that Shannon-type information inequalities are in general not sufficient for computing network coding capacities. To accomplish this, we obtain a capacity upper bound for the Vamos network using a non-Shannon-type information inequality discovered in 1998 by Zhang and Yeung, and then show that it is smaller than any such bound derived from Shannon-type information inequalities. This is the first application of a non-Shannon-type inequality to network coding. We also compute the exact routing capacity and linear coding capacity of the Vamos network. Finally, using a variation of the Vamos network, we prove that Shannon-type information inequalities are insufficient even for computing network coding capacities of multiple-unicast networks.

All unconstrained information inequalities in three or fewer random variables are known to be "Shannon-type", in that they are nonnegative linear combinations of instances of the inequality I(A;B|C) ges 0. In 1998, Zhang and Yeung gave the first example of an information inequality in four variables that is not "Shannon-type". Here we give six new unconstrained non-Shannon information inequalities in four variables. The new inequalities are independent of each other and of the Zhang-Yeung inequality

We define the routing capacity of a network to be the supremum of all possible fractional message throughputs achievable by routing. We prove that the routing capacity of every network is achievable and rational, we present an algorithm for its computation, and we prove that every rational number in (0, 1] is the routing capacity of some solvable network. We also determine the routing capacity for various example networks. Finally, we discuss the extension of routing capacity to fractional coding solutions and show that the coding capacity of a network is independent of the alphabet used

The coding capacity of a network is the supremum of ratios k/n, for which there exists a fractional (k,n) coding solution, where k is the source message dimension and n is the maximum edge dimension. The coding capacity is referred to as routing capacity in the case when only routing is allowed. A network is said to achieve its capacity if there is some fractional (k,n) solution for which k/n equals the capacity. The routing capacity is known to be achievable for arbitrary networks. We give an example of a network whose coding capacity (which is 1) cannot be achieved by a network code. We do this by constructing two networks, one of which is solvable if and only if the alphabet size is odd, and the other of which is solvable if and only if the alphabet size is a power of 2. No linearity assumptions are made.

We prove that for any finite directed acyclic network, there exists a corresponding multiple-unicast network, such that for every alphabet, each network is solvable if and only if the other is solvable, and, for every finite-field alphabet, each network is linearly solvable if and only if the other is linearly solvable. The proof is constructive and creates an extension of the original network by adding exactly s+5m(r-1) new nodes where, in the original network, m is the number of messages, r is the average number of receiver nodes demanding each source message, and s is the number of messages emitted by more than one source. The construction is then used to create a solvable multiple-unicast network which becomes unsolvable over every alphabet size if all of its edge directions are reversed and if the roles of source-receiver pairs are reversed

We define the routing capacity of a network to be the supremum of all possible fractional message throughputs achievable by routing. We prove that the routing capacity of every network is achievable and rational, we present an algorithm for its computation, and we prove that every non-negative rational number is the routing capacity of some network. We also determine the routing capacity for various example networks. Finally, we discuss the extension of routing capacity to fractional coding solutions and show that the coding capacity of a network is independent of the alphabet used

It is known that every solvable multicast network has a scalar linear solution over a sufficiently large finite field alphabet. It is also known that this result does not generalize to arbitrary networks. There are several examples in the literature of solvable networks with no scalar linear solution over any finite field. However, each example has a linear solution for some vector dimension greater than one. It has been conjectured that every solvable network has a linear solution over some finite field alphabet and some vector dimension. We provide a counterexample to this conjecture. We also show that if a network has no linear solution over any finite field, then it has no linear solution over any finite commutative ring with identity. Our counterexample network has no linear solution even in the more general algebraic context of modules, which includes as special cases all finite rings and Abelian groups. Furthermore, we show that the network coding capacity of this network is strictly greater than the maximum linear coding capacity over any finite field (exactly 10% greater), so the network is not even asymptotically linearly solvable. It follows that, even for more general versions of linearity such as convolutional coding, filter-bank coding, or linear time sharing, the network has no linear solution

It is known that every solvable multicast network has a scalar linear solution over a sufficiently large finite-field alphabet. It is also known that this result does not generalize to arbitrary networks. There are several examples in the literature of solvable networks with no scalar linear solution over any finite field. However, each example has a linear solution for some vector dimension greater than one. It has been conjectured that every solvable network has a linear solution over some finite-field alphabet and some vector dimension. We provide a counterexample to this conjecture. We also show that if a network has no linear solution over any finite field, then it has no linear solution over any finite commutative ring with identity. Our counterexample network has no linear solution even in the more general algebraic context of modules, which includes as special cases all finite rings and Abelian groups. Furthermore, we show that the network coding capacity of this network is strictly greater than the maximum linear coding capacity over any finite field (exactly 10% greater), so the network is not even asymptotically linearly solvable. It follows that, even for more general versions of linearity such as convolutional coding, filter-bank coding, or linear time sharing, the network has no linear solution.

We consider the degree-diameter problem for Cayley graphs of Abelian groups (Abelian graphs) for both directed graphs and undirected graphs. The problem is closely related to that of finding efficient lattice coverings of Euclidean space by shapes such as octahedra and tetrahedra; we exploit this relationship in both directions. For two generators (dimensions), these methods yield optimal Abelian graphs with a given diameter k. (The results in two dimensions are not new; they are given in the literature of distributed loop networks.) We find an undirected Abelian graph with three generators and a given diameter k, which we conjecture to be as large as possible; for the directed case, we obtain partial results. These results are connected to efficient lattice coverings of ${\bf R}^3$ by octahedra or by tetrahedra; computations on Cayley graphs lead us to such lattice coverings, which we conjecture to be optimal. (The problem of finding such optimal coverings can be reduced to a finite number of nonlinear optimization problems.) We discuss the asymptotic behavior of the Abelian degree-diameter problem for large numbers of generators. The graphs obtained here are substantially better than traditional toroidal meshes, but, in the simpler undirected cases, retain certain desirable features such as good routing algorithms, easy constructibility, and the ability to host mesh-connected numerical algorithms without any increase in communication times.

It is known that for every solvable multicast network, there exists a large enough finite-field alphabet such that a scalar linear solution exists. We prove: i) every binary solvable multicast network with at most two messages has a binary scalar linear solution; ii) for more than two messages, not every binary solvable multicast network has a binary scalar linear solution; iii) a multicast network that has a solution for a given alphabet might not have a solution for all larger alphabets.

Kantorovich and Livenson [6] initiated the study of infinitary Boolean operations applied to the subsets of the Baire space and related spaces. It turns out that a number of interesting collections of subsets of the Baire space, such as the collection of Borel sets of a given type (e.g. the Fσ sets) or the collection of analytic sets, can be expressed as the range of an ω-ary Boolean operation applied to all possible ω-sequences of clopen sets. (Such collections are called clopen-ω-Boolean.) More recently, the ranges of I-ary Boolean operations for uncountable I have been considered; specific questions include whether the collection of Borel sets, or the collection of sets at finite levels in the Borel hierarchy, is clopen-I-Boolean. The main purpose of this paper is to give a characterization of those collections of subsets of the Baire space (or similar spaces) that are clopen-I-Boolean for some I. The Baire space version can be stated as follows: a collection of subsets of the Baire space is clopen-I-Boolean for some I iff it is nonempty and closed downward and σ-directed upward under Wadge reducibility, and in this case we may take I = ω2. The basic method of proof is to use discrete subsets of spaces of the form K2 to put a number of smaller clopen-I-Boolean classes together to form a large one. The final section of the paper gives converse results indicating that, at least in some cases, ω2 cannot be replaced by a smaller index set.

A subset of the Cantor space ω2 is called monotone iff it is closed upward under the partial ordering ≤ defined by x ≤ y iff x(n) ≤ y(n) for all n ∈ ω. A set is -positive ( -positive) iff it is monotone and -positive set is a countable union of -positive sets; a -positive set is a countable intersection of -positive sets. (See Cenzer [2] for background information on these concepts.) It is clear that any -positive set is and monotone; the converse holds for n ≤ 2 [2] and was conjectured by Dyck to hold for greater n. In this note, we will disprove this conjecture by giving examples of monotone sets (for n ≥ 3) which are not even -positive. First we note a few isomorphisms. The space (ω2, ≤) is isomorphic to the space (ω2 ≥), so instead of monotone and positive sets we may construct hereditary and negative sets (the analogous notions with “closed upward” replaced by “closed downward”). Also, (ω2, ≤) is isomorphic to ( (ω), ⊆), where denotes the power set operator, or to ( (S), ⊆) for any countably infinite set S. In order to remove extraneous notation from the proofs, we state the results in an abstract form (whose generality is deceptive).